Question

question 1
v1 = 84 cu/ft
p1 = 300 psi
t1 = 48°c
v2 =????? cu/ft
p2 = 110 psi
t2 = 30°c

Explanation:

Step1: Convert temperatures to Kelvin

To use the ideal gas law (combined gas law), we need to convert Celsius to Kelvin. The formula is \( T(K) = T(^\circ C) + 273.15 \).
For \( T_1 \): \( T_1 = 48 + 273.15 = 321.15 \, K \)
For \( T_2 \): \( T_2 = 30 + 273.15 = 303.15 \, K \)

Step2: Apply the combined gas law

The combined gas law is \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). We solve for \( V_2 \):
\( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \)

Substitute the values: \( P_1 = 300 \, psi \), \( V_1 = 84 \, cu/ft \), \( T_2 = 303.15 \, K \), \( P_2 = 110 \, psi \), \( T_1 = 321.15 \, K \)

\( V_2 = \frac{300 \times 84 \times 303.15}{110 \times 321.15} \)

First, calculate numerator: \( 300 \times 84 = 25200 \); \( 25200 \times 303.15 = 25200 \times 300 + 25200 \times 3.15 = 7560000 + 79380 = 7639380 \)

Denominator: \( 110 \times 321.15 = 35326.5 \)

Then, \( V_2 = \frac{7639380}{35326.5} \approx 216.25 \, cu/ft \) (Wait, let's recalculate more accurately. Let's simplify the fractions first. \( \frac{300}{110} = \frac{30}{11} \), \( \frac{303.15}{321.15} \approx \frac{303}{321} = \frac{101}{107} \) (approx). Then \( V_2 = 84 \times \frac{30}{11} \times \frac{303.15}{321.15} \). Let's compute \( \frac{30 \times 303.15}{11 \times 321.15} \approx \frac{9094.5}{3532.65} \approx 2.574 \). Then \( 84 \times 2.574 \approx 216.2 \). Wait, but maybe I made a miscalculation. Wait, let's do it step by step:

\( 300 \times 84 = 25200 \)
\( 25200 \times 303.15 = 25200 \times 303.15 \). Let's compute 25200 300 = 7,560,000; 25200 3.15 = 79,380; total 7,639,380.

Denominator: 110 * 321.15 = 35,326.5

Now divide 7,639,380 by 35,326.5: 35,326.5 216 = 35,326.5 200 + 35,326.5 16 = 7,065,300 + 565,224 = 7,630,524. Subtract from numerator: 7,639,380 - 7,630,524 = 8,856. Then 8,856 / 35,326.5 ≈ 0.25. So total ≈ 216.25 cu/ft. Wait, but that seems high. Wait, maybe the units? Wait, psi is pressure, cu/ft is volume. Wait, maybe I messed up the combined gas law. Wait, no, the combined gas law is correct for constant amount of gas. Wait, let's check the numbers again. P1 is 300 psi, P2 is 110 psi (lower pressure), T1 is 48C, T2 is 30C (lower temperature). So volume should increase with lower pressure and decrease with lower temperature. Let's see: P decreases by a factor of 300/110 ≈ 2.727, T decreases by 303/321 ≈ 0.944. So overall, V2 = 84 (300/110) (303/321) ≈ 84 2.727 0.944 ≈ 84 2.57 ≈ 216. So that's correct.

Answer:

\( \approx 216 \, cu/ft \) (more precisely, ~216.25 cu/ft)